Find the linearizations L(x, y, z) of the functions at the given points. f(x, y, z)=x y+y z+x z

Find the linearizations L(x, y, z) of the functions at the...

Key Concepts

First-Order Taylor Expansion

The first-order Taylor expansion is a method to approximate a function using its value and its first derivatives at a specific point. In the context of functions of several variables, it gives an affine approximation (or the tangent plane) that estimates the function's behavior near the point of expansion, simplifying analysis and computation.

Linearization

Linearization is the process of approximating a nonlinear function by a linear function near a given point, typically through the use of first-order Taylor expansion. This approximation, often representing the tangent plane at the point of tangency, provides a simple model to analyze the behavior of the function close to that point.

Partial Derivatives

Partial derivatives measure the rate at which a multivariable function changes with respect to one variable, while keeping the other variables constant. They are used to capture the sensitivity of the function’s output to changes in one of its inputs, and are crucial to understanding the local behavior of functions of several variables.

Multivariable Functions

Multivariable functions are functions that depend on two or more independent variables. They are studied in calculus to understand how changes in several input variables affect the output. This area of study lays the foundation for analyzing surfaces and higher dimensional objects, and it is key to many applications in physics, economics, engineering, and other fields.

The Gradient

The gradient of a multivariable function is a vector consisting of all its partial derivatives. This vector points in the direction of the steepest ascent of the function, and its magnitude corresponds to the rate of maximum increase. The gradient is used in many areas such as optimization and in forming the linear approximation of a function.

Transcript

00:02 We're given the function f of x, y, z equals x times y plus y x times z plus x times z.

00:19 And we want to find the linearization of this function at a couple of different points that we are given.

00:26 So in order to find the linearization, that linearization is the value of the function at our point x0, y, 0, z0, plus the partial derivative with respect to x, evaluated at our point, times x minus x0, plus the partial derivative with respect to y, evaluated at the point x0, x0 y0 z0 times y minus y0 plus the partial derivative with respect to z evaluated at the point x0 y0 z 0 times the quantity x z minus z zero so since we are only going to be using different points in each part of these problems let's just find the linearization formula first and then all we have to do is plug in our values for the coordinates.

01:44 So our linearization is going to be the xy plus yz plus xz, our function, plus the partial derivative with respect to x, so that means treating y and z as constants.

02:05 So the partial derivative of xy will just be y.

02:10 Partial derivative of y z will be zero because that's considered as one entire constant, and then the partial derivative of xz will be plus z, and that'll be multiplied times x minus our x coordinate of the point that we're given.

02:30 Now we need the partial derivative with respect to y.

02:33 So similarly treating x and z as constants, we end up with x plus z times the quantity, y minus y sub 0 and then plus now the partial derivative with respect to z so x and y are constants so when we do that we end up with y plus x times z minus z sub 0 so that's the formula for our linearization and now it's a matter of fixing my notation and then plugging in our coordinates that should be l of x, y, and z...

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